Block Diagram Modeling Linearly-Implicit ODEs
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Solution Overview
Problem
Existing block diagram modeling environments lack the capability to effectively model and execute systems described in a linearly-implicit formulation of ordinary differential equations (ODEs) or differential algebraic equations (DAEs), particularly in simulating physical systems with algebraic loops.
Innovation Solution
The development of methods and solvers within block diagram modeling environments to model and execute systems in a linearly-implicit formulation of ODEs or DAEs, allowing for the creation and execution of models using user-defined blocks and exploiting sparsity structures in mass matrices, enabling the simulation of systems with algebraic loops.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If block diagram modeling environments use traditional explicit formulation of ODEs, then the modeling is straightforward and easy to implement, but they cannot effectively model and execute systems described in linearly-implicit formulation of ODEs or DAEs
Solution Approach 1:
The patent introduces an intermediary mass matrix M(t,x,y) that bridges the explicit formulation y'=f(t,x,y) and the linearly-implicit formulation M(t,x,y)y'=f(t,x,y). This mass matrix acts as a mediator that enables the block diagram environment to handle linearly-implicit systems while maintaining the familiar explicit formulation interface, thus resolving the contradiction between adaptability and complexity.
Solution Approach 2:
The patent transforms the system formulation by introducing the mass matrix parameter M(t,x,y), which changes the mathematical structure from explicit to linearly-implicit formulation. This parameter change enables the modeling of a broader class of systems (including DAEs with algebraic loops) while the block diagram environment absorbs the increased mathematical complexity through automated solver integration.
2Adaptability or versatility
If block diagram environments support linearly-implicit formulation of ODEs or DAEs, then they can simulate a broader range of physical systems including those with algebraic loops, but the solving complexity and computational requirements increase
Solution Approach 1:
The patent segments the solution process by separating the mass matrix evaluation M(t,x,y) from the forcing function evaluation f(t,x,y), allowing specialized numerical solvers to handle each component efficiently. This segmentation enables the use of optimized algorithms for linearly-implicit systems while maintaining simulation efficiency through modular computation.
Solution Approach 2:
The patent leverages existing numerical solver algorithms designed for linearly-implicit ODEs and DAEs, copying and adapting proven computational methods into the block diagram environment. This approach enables accurate simulation of complex physical systems with algebraic loops while avoiding the need to develop new solving algorithms from scratch, thus maintaining productivity.
3Reliability
If traditional solvers are used in block diagram environments, then the execution is simple and fast, but they lack the capability to solve linearly-implicit formulation of ODEs or DAEs
Solution Approach 1:
The patent implements a universal solver framework that can handle both traditional explicit ODEs and linearly-implicit ODEs/DAEs through a unified interface. The solver automatically detects the system type and applies appropriate algorithms, providing reliable solutions for linearly-implicit systems while maintaining simplicity for standard cases. This multi-functionality resolves the contradiction by making solver complexity transparent to the user.
Data Source
AI summary
Methods for modeling and executing systems in a linearly-implicit formulation of ordinary differential equations (ODEs) or differential algebraic equations (DAEs) are provided in modeling environments. The modeling environments may include block diagram modeling environments, such as time-based block diagram modeling environments. The block diagram modeling environments may include or be coupled to other modeling environments, such as physical modeling environments, so that models created in other modeling environments may also be introduced and solved in the linearly-implicit formulation of ODEs or DAEs in the block diagram modeling environments. Models describing the systems in the linearly-implicit formulation of ODEs or DAEs may also be created directly by users using user-defined blocks in the block diagram modeling environments. The present invention provides solvers for solving the system described in the linearly-implicit formulation of DAEs as well as ODEs in the block diagram modeling environment.